Theorem zringring

Description: The ring of integers is a ring. (Contributed by AV, 20-May-2019) (Revised by AV, 9-Jun-2019) (Proof shortened by AV, 13-Jun-2019)

	Ref	Expression
Assertion	zringring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		zringcrng	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{CRing}$$
2		crngring	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{CRing}\to {\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{Ring}$$
3	1 2	ax-mp	$$⊢{\mathbb{Z}}_{\mathrm{ring}}\in \mathrm{Ring}$$